Maximal Ideal of rings

What does the maximal ideal of rings means? Though I know the meaning of ideal, I just couldnot figure out What does the maximal ideal of rings means? If possible please explain me the meaning of quotient ring as well.

What does the maximal ideal of rings means? Though I know the meaning of ideal, I just couldnot figure out What does the maximal ideal of rings means? If possible please explain me the meaning of quotient ring as well.

A maximal ideal can be thought of as the maximal element of the partially ordered set . It's really just an ideal such that if is an ideal with then . It's the 'biggest' ideal.

A quotient ring is just like all the other quotient structures in algebra. Given a two-sided ideal we can partition into equivalence classes where the equivalence relation is and we usually denote by the coset notation . We can then define a ring structure on by and .

To show that's well-defined isn't obvious, and for anything more you'll need to see a book. I suggest Dummit and Foote or Herstein.

I thought this was clear(ish) by my comment about being maximal elements in the poset.

Sure, but you still need to mention Zorn's lemma. It is standard to assume maximal ideals exist, but you still need assume it. The words Zorn, choice, well-ordered or assume didn't appear in your post...